We show that the application of atom interferometry techniques to the internal, i.e., rotational-vibrational states of molecules provides a new tool for ultrahigh precision tests of fundamental physics. The measurement principle is based on the fact that the electronic structure of molecules is not spherically symmetric. A diatomic quantum sensor can hence distinguish between the direction along its internuclear axis and the two orthogonal directions and is therefore direction sensitive. As an example we show how a molecular rotational-vibrational quantum interferometer based on the hydrogen deuteride molecular ion
(HD+) may be used to detect gravitational waves. We show that a monochromatic gravitational wave of dimensionless amplitude $h={10}^{\ensuremath{-}19}$ will cause a frequency shift of the order of $30\text{ }\ensuremath{\mu}\text{Hz}$ between appropriately prepared quantum states, a frequency difference likely to be detectable with the next generation atom interferometers in 1 s.
Molecules can be used to test fundamental physics. Such tests often require cold molecules for detailed spectroscopic analysis. Cooling internal degrees of freedom provides a high level of state-selectivity, with large populations in the molecular states of interest. Cold translational motion allows slow, bright beams to be created, allowing long interaction times. In this tutorial article we describe the common techniques for producing cold molecules for high-precision spectroscopy experiments. For each technique we give examples of its application in experiments that use molecular structure to probe fundamental physics, choosing one experiment in particular as a case study. We then discuss a number of new techniques, some currently under development, others proposed, that promise high flux sources of cold molecules applicable to precise spectroscopic tests of fundamental physics.